I understand that a percentage is saying, how many parts there are out of 100. So 65% represents 65 parts of out 100 possible parts.
One thing that I can't get my head around, or I am over thinking, is that if we have something such as $9/12$ and I wish to know what percentage those 9 parts out of the 12 take up, what exactly am I doing? Am I saying that if the 12 parts were infact 100 parts, how much would the 9 parts now take up?
We have $\frac{9}{12}=\frac{3}{4}=\frac{75}{100}$, which is evidently $75%$. This means that for every 12 objects, you get 9 objects, or put more simply, for every 4 objects you get 3 objects.
In general, given any fraction, to find its percentage, multiply the fraction by $100%$.
Here’s an easier way to approach percentages:
The sign $\%$ is similar to symbols like $e$ (Euler’s number) and $\rm m$ (the metre)—it simply stands in for another quantity—in this case, $\frac1{100}$. To solve, all you need to do is carry out the multiplication.
$$\text{$P\%$ of $X$} = (P\%)(X) = P\cdot \frac1{100}\cdot X = \frac{PX}{100}$$
You probably know that, for example, $27\%=0.27$, and this is why.
To turn $\frac9{12}$ into a percentage, first divide $9$ by $12$:
$$\frac{9}{12}=0.75$$
Now, move the decimal point to the right two spaces (i.e., multiply the quotient by $100$) and then tack on the percentage sign (i.e., multiply the quotient by $\frac1{100}$):
$$\frac9{12}=0.75=75\%$$
Working from a percentage in reverse however, is not so simple. Because $$\text{percentage}={ \text{# parts} \over \text{# whole} }$$ we can never tell how many “parts” there are or what the size of the “whole” group is without more information.
For example, which country spends more on education: the United States or Costa Rica? The US spends only $4.9\%$ of its GDP, while CR spends a much greater $7.6\%$. However, the question asks you to compare actual expenditures, so percentages do not give you the entire story.
The GDP of the US is \$19.36 trillion ($ \$ 1.94\times10^{13}$) while the GDP of CR is only \$85.2 billion ($ \$ 8.52\times10^{10}$). Therefore, each country spends
$$\begin{array}{cccl} \text{US:} & (4.9\%)\left(\$ 1.94\times10^{13}\right) & = & \$9.506\times10^{11} \\ \text{CR:} & (7.6\%)\left(\$ 8.52\times10^{10}\right) & = & \$6.475\times10^{9} \\ \end{array}$$
As you can see, the percentages belie which country spends more. The moral of the story: percentages disguise the number of parts as well as the number in the whole sample. There is just no way to tell how many parts you’re dealing with without having more information.
Note, however, that sometimes we want this, because it facilitates comparisons across groups.
The choice of 100 is rather arbitrary. If we had six fingers on each hand, we'd probably use pergrossages rather than percentages. Then your 9 twelfths would be 108 pergrossage.
Perhaps the most natural thing is parts of a unit, that is, parts of 1. One divided by 12 is approximately 0.083333, periodic but non-terminating in decimal. Multiply that by 9 and you should get 0.75, assuming no loss in calculating precision.
If that looks like 75%, it's only because we use the decimal system. Try this in Wolfram Alpha: 3/4 in duodecimal Then try 108 in duodecimal
$$\frac{9}{12}=\frac{3}{4}=\frac{3}{4}\frac{25}{25}=\frac{75}{100}$$
If we have something such as $\frac9{12}$ and I wish to know what percentage those $9$ parts out of the $12$ take up, what am I doing?
Don't worry. We just need to find: $$\frac9{12}\times 100 = 75\%$$ And yes, this is an another way of saying that if $12$ corresponds to a $100$, then $9$ corresponds to $75$.
